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An example of how the Ricci flow can increase topological entropy

Published online by Cambridge University Press:  01 December 2007

DAN JANE
DAN JANE*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (email: jdtj2@cam.ac.uk)
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Abstract

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We give a surface for which the Ricci flow applied to the metric will increase the topological entropy of the geodesic flow. Specifically, we first adapt the Melnikov method to apply to a Ricci flow perturbation and then we construct a surface which is closely related to a surface of revolution, but does not quite have rotational symmetry. This is done by adapting the Liouville metric representation of a surface of revolution. The final steps of the Melnikov method require numerical integration.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 6 , December 2007, pp. 1919 - 1932
Copyright
Copyright © Cambridge University Press 2007

References

[1]Bolsinov, A. V. and Fomenko, A. T.. Integrable Hamiltonian Systems: Geometry, Topology, Classification. Chapman & Hall/CRC, Boca Raton, FL, 2004.CrossRefGoogle Scholar
[2]Brin, M. and Stuck, G.. Introduction to Dynamical Systems. Cambridge University Press, Cambridge, 2002.CrossRefGoogle Scholar
[3]Burns, K. and Weiss, H.. A geometric criterion for positive topological entropy. Comm. Math. Phys. 172 (1995), 95–118.CrossRefGoogle Scholar
[4]Chow, B.. The Ricci flow on the 2-sphere. J. Differential Geom. 33 (1991), 325–334.CrossRefGoogle Scholar
[5]Chow, B. and Knopf, D.. The Ricci Flow: An Introduction. American Mathematical Society, Providence, RI, 2004.CrossRefGoogle Scholar
[6]Darboux, G.. Leçons sur la théorie générale des surfaces et les applications géometriques du calcul infinitésimal. Gauthier-Villars, Paris, 1894.Google Scholar
[7]Hamilton, R.. The Ricci flow on surfaces. Comm. Math. Phys. 138 (1991), 19–31.Google Scholar
[8]Hamilton, R.. The Formation of Singularities in the Ricci Flow. International Press, Somerville, MA, 1995.Google Scholar
[9]Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.CrossRefGoogle Scholar
[10]Knieper, G., Katok, A. and Weiss, H.. Formulas for the derivatives and critical points of topological entropy for Anosov and geodesic flows. Comm. Math. Phys. 138 (1991), 19–31.Google Scholar
[11]Manning, A.. The volume entropy of a surface decreases along the Ricci flow. Ergod. Th. & Dynam. Sys. 24 (2004), 171–176.CrossRefGoogle Scholar
[12]Melnikov, V. K.. On the stability of the centre for time periodic perturbations. Trans. Moscow Math. Soc. 12 (1963), 1–57.Google Scholar
[13]Morgan, J. and Tian, G.. Ricci Flow and the Poincaré Conjecture (Clay Mathematics Monographs, 3). CMI/American Mathematical Society, Providence, RI, 2007.Google Scholar
[14]Paternain, G.. Geodesic Flows. Birkhäuser, Basel, 1999.CrossRefGoogle Scholar
[15]Robinson, C.. Horseshoes for autonomous Hamiltonian systems using the Melnikov method. Ergod. Th. & Dynam. Sys. 8 (1988), 395–409.Google Scholar
[16]Wiggins, S.. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York, 1988.Google Scholar